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Full text of "Treatise On Analysis Vol-Ii"

136 XIII INTEGRATION Finally, we have to show that (b) implies (a). The space X is the union of an increasing sequence (HM) of compact sets (3.18.3). For each n we use the hypothesis (b) together with (13.7.9) to define a sequence (K.mn)m^1 of com- pact sets such that Kln cz Hn ~~ HH-l9 Kmn c (Hn - H^J - (J Kin, and and such that the restrictions u \ Kmn are continuous. The complement of the union of the sets Kmn for n ^ 1 and m ^ 1 is then negligible, and consequently u is measurable. It follows that, for a mapping/: X -»• R to be measurable, it is necessary and sufficient that, for each compact subset K of X, the function/^ (which by convention is taken to be zero at all points of X — K, even if /is infinite at such points (cf. (13.11)) should be measurable. For this condition is sufficient by virtue of (13.9.4), and it is necessary by (13.9.4) and the fact that X — K is measurable (13.9.3). The same reasoning shows that if /is a mapping of X into R and K is a compact subset of X such that f\ K is continuous, then fcpK is measurable. (13.9,5) Let (YJ be a sequence of topo logical spaces, and for each n let un be a measurable mapping of X into Yn . Then, given any compact subset K of X and any & > 0, there exists a compact subset K' of K such that ju(K — K') ^ e and such that un \ K' is continuous for all n. Using (13.9.4) we define inductively a decreasing sequence (KJ of compact subsets of K such that ^(K - KJ g & and XKn - Kn+1) <£ e/2"+I, and such that un | Kn is continuous. Then by (13.8.7) it is clear that K' = f] Kn satisfies n the required conditions. (1 3.9.6) Let (Y^jL ^t^nbe a finite sequence of topological spaces, Z a topological space, and v a continuous mapping ofY\ Yf into Z. Let ui : X -» Yf (1 ^ / ^ n) be measurable. Then the mapping x^~^v(u1(x), . . . , un(x)) is measurable. This follows immediately from (13.9.5) and (13.9.4). (13.9.7) Iff, g are two measurable functions with values in R, then sup(/ g) andinf(f, g) are measurable. Ifu,v are measurable mappings ofX into a (real or complex) vector space, then u + v and au (where a is any scalar) are measur- able.ch of